Sparse-Wire Diagnosis of Hinge-Localized States in Wannier Tight-Binding Hamiltonians

<jats:p>Hinge states are a characteristic boundary manifestation of three-dimensional higherorder topological phases, but identifying them in realistic Wannier tight-binding models requires a sequence of choices that is rarely automated: selection of a target bulk gap, construction of multiple mixed-boundary geometries, sparse solution of large wire Hamiltonians, and discrimination of corner-localized spectral weight from surface and bulk weight. We present HingeStateDetector, a computational method that performs this sequence using a wannier90_hr.dat Hamiltonian and a POSCAR as its mandatory inputs. The program parses and validates the real-space hopping model, estimates sampled global bulk gaps, constructs wires that are finite along two lattice directions and periodic along the third, and uses shift--invert sparse diagonalization near the selected gap. For every eigenstate it reports corner-, edge-, and interior-cell probabilities and builds an energy- and cell-resolved local density of states (LDOS). An automatic mode screens all three lattice axes, while machinereadable JSON and NPZ outputs retain the parameters and numerical evidence needed for convergence studies. We verify the implementation with the chiral hinge model of Benalcazar, Bernevig, and Hughes: only the expected periodic direction is classified as positive, four corner-localized branches traverse the bulk gap, and the interior weight falls below one percent for converged cross sections. A 30-orbital spinful Bi$_2$Se$_3$ Hamiltonian distributed with WannierTools provides a realistic stress test. In addition, a 128-orbital spinful Hamiltonian for the experimentally established higher-order topological insulator $\alpha$-Bi$_4$Br$_4$ supplies a real-material positive control. The detector selects the quasi-one-dimensional chain direction and retains corner-dominated in-gap candidates as the cross section grows from $3\times3$ to $5\times5$. These results establish an axis-resolved, quantitative route from Wannier interpolation to finite-boundary evidence, while showing explicitly why hinge localization alone is not a bulk topological invariant</jats:p>